Sherman–Morrison formula

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In mathematics, in particular linear algebra, the Sherman–Morrison formula,[1][2][3] named after Jack Sherman and Winifred J. Morrison, computes the inverse of the sum of an invertible matrix and the outer product, , of vectors and . The Sherman–Morrison formula is a special case of the Woodbury formula. Though named after Sherman and Morrison, it appeared already in earlier publications.[4]


Suppose is an invertible square matrix and , are vectors. Suppose furthermore that . Then the Sherman–Morrison formula states that

Here, is the outer product of two vectors and . The general form shown here is the one published by Bartlett.[5]


If the inverse of is already known, the formula provides a numerically cheap way to compute the inverse of corrected by the matrix (depending on the point of view, the correction may be seen as a perturbation or as a rank-1 update). The computation is relatively cheap because the inverse of does not have to be computed from scratch (which in general is expensive), but can be computed by correcting (or perturbing) .

Using unit columns (columns from the identity matrix) for or , individual columns or rows of may be manipulated and a correspondingly updated inverse computed relatively cheaply in this way.[6] In the general case, where is a times matrix and and are arbitrary vectors of dimension , the whole matrix is updated[5] and the computation takes scalar multiplications.[7] If is a unit column, the computation takes only scalar multiplications. The same goes if is a unit column. If both and are unit columns, the computation takes only scalar multiplications.


We verify the properties of the inverse. A matrix (in this case the right-hand side of the Sherman–Morrison formula) is the inverse of a matrix (in this case ) if and only if .

We first verify that the right hand side () satisfies .

Note that is a scalar, so can be factored out, leading to:

In the same way, it is verified that

Following is an alternate verification of the Sherman–Morrison formula using the easily verifiable identity

Let and , then

Substituting gives

See also


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  6. Langville, Amy N.; and Meyer, Carl D.; "Google's PageRank and Beyond: The Science of Search Engine Rankings", Princeton University Press, 2006, p. 156
  7. Update of the inverse matrix by the Sherman–Morrison formula

External links